Vedran Dizdarevic
24. May 2006
Bayesian Methods in Positioning Applications – p.2/21
GRAZ UNIVERSITY OF TECHNOLOGY
Advanced Signal Processing Seminar
Problem Statement (1/2)
System description in two steps Find system model (modeling) Find the particular realization of the model
(identification; parameter estimation θ)
Time variant system parameters require recursive estimation on-line vs. off-line data processing
Bayesian Methods in Positioning Applications – p.3/21
GRAZ UNIVERSITY OF TECHNOLOGY
Advanced Signal Processing Seminar
With N-dimensional observation zk = {z1 k, . . . , z
N k }
Initial condition(s) - selection of θ0
Bayesian Methods in Positioning Applications – p.4/21
GRAZ UNIVERSITY OF TECHNOLOGY
Advanced Signal Processing Seminar
RLS
systems Extension of linear regression (LS) methods
Bayesian inference Estimation of distribution of θ
Bayesian Methods in Positioning Applications – p.5/21
GRAZ UNIVERSITY OF TECHNOLOGY
Advanced Signal Processing Seminar
Introduction to Bayesian Estimation
variable with p(θ)
of p(θ)
Relying on a Bayes rule updated p(θ) isinferredd from prior (predicted) p(θ) combined with incoming observations
Bayesian Methods in Positioning Applications – p.6/21
GRAZ UNIVERSITY OF TECHNOLOGY
Advanced Signal Processing Seminar
Bayes Rule and Contemporary Science
Published in T. Bayes Essay Towards Solving a Problem in the Doctrine of Chances (1764)
Steam engine of J. Watt (1769)
B. Franklin defines the concept of positive and negative polarity (1752)
L. Galvani and frogs (1770)
K.F. Gauss was born (1777)
Bayesian Methods in Positioning Applications – p.7/21
GRAZ UNIVERSITY OF TECHNOLOGY
Advanced Signal Processing Seminar
Bayes Rule
Assume a disjoint set of n discrete events {Ei} n i=1
Probabilities P (Ei) n i=1 are assigned to each event
If events are not independent observing that event Ek
hasoccurredd, probabilities of all other events {Ei} n i=1
will change
This is described using conditional probability P (Ei|Ek)
P (Ei, Ek) = P (Ei|Ek)P (Ek) = P (Ek|Ei)P (Ei)
P (Ei|Ek) = P (Ek|Ei)P (Ei) P (Ek)
Bayesian Methods in Positioning Applications – p.8/21
GRAZ UNIVERSITY OF TECHNOLOGY
Advanced Signal Processing Seminar
Motion model θk+1 = f(θk) + wk
θk,wk ∈ R d, f : R
d → R d
d → R mk
GRAZ UNIVERSITY OF TECHNOLOGY
Advanced Signal Processing Seminar
Likelihood function Lk(θk) = p(zk|θk) = pvk
(zk − hk(θk))
(θk+1 − f(θk))
GRAZ UNIVERSITY OF TECHNOLOGY
Advanced Signal Processing Seminar
Select initial distribution p0|0 and set k = 1
Step 1 Compute the predictive pdf pk|k−1(θk) =
∫ Rd p(θk|θk−1)p(θk−1|z1, . . . , zk−1)dθ
Step 2 Compute the posterior pdf pk|k(θk) = p(zk|θk)p(θk|z1,...,zk−1)
p(zk|z1,...,zk−1) ∝ pk|k−1(θk)Lk(θ)
Step 3 Output θk and variance V (pk|k(θk))
Step 4 Increment k and repeat from Step 1
Bayesian Methods in Positioning Applications – p.11/21
GRAZ UNIVERSITY OF TECHNOLOGY
Advanced Signal Processing Seminar
θk = argmax(pk|k(θk)) (MAP)
Problem: it is difficult to find an analytical solution to this problem!
If the noise pdfs are independent, white and zero-mean Gaussian and the measurement equation linear in θ, it can be derived that the Bayesian approach reduces to the Kalman Filter
A general algorithm which approximates the pdfs with arbitrary accuracy is the particle filter
Bayesian Methods in Positioning Applications – p.12/21
GRAZ UNIVERSITY OF TECHNOLOGY
Advanced Signal Processing Seminar
Numerical means for the tracking the evolution of the pdfs
The pdf is approximated as weighted sum of samples in the space state (particles) p(θ) ≈
∑M m=1 w(m)δ(x − x(m))
Recursive update Prediction: Calculate the predictive pdf using the
motion model Update: Weights update with normalization
Bayesian Methods in Positioning Applications – p.13/21
GRAZ UNIVERSITY OF TECHNOLOGY
Advanced Signal Processing Seminar
Particle Filter (2/4)
Initialization Select an importance function π and create a set of
particles x (m) 0 ∼ π
Measurement update Weight update according to the likelihood
w (m) k = w
(m) k−1π(θ|z)
(m) k /
∑ m w
(m) k
m=1 w (m) k θ
(m) k
Bayesian Methods in Positioning Applications – p.14/21
GRAZ UNIVERSITY OF TECHNOLOGY
Advanced Signal Processing Seminar
Weights tend to concentrate in few particles
Draw P particles from the current particle with probabilities proportional to the corresponding weight; set weights for the new particle as wk = 1
P
Optionally resampling only if a given criterion is fulfilled Neff = 1
P
GRAZ UNIVERSITY OF TECHNOLOGY
Advanced Signal Processing Seminar
GRAZ UNIVERSITY OF TECHNOLOGY
Advanced Signal Processing Seminar
1+θ2 k−1
GRAZ UNIVERSITY OF TECHNOLOGY
Advanced Signal Processing Seminar
Pro and Contra This algorithm is known under many names!
Pro: Non-linear sate model needs no approximations Non-Gaussian distributions θ
Combines different observations in a single model Variable dimension of the observation vector no problem
Contra: Computationally expensive Selection of prior distribution
Bayesian Methods in Positioning Applications – p.18/21
GRAZ UNIVERSITY OF TECHNOLOGY
Advanced Signal Processing Seminar
Application: Positioning, Navigation, Tracking
Observations: z = [pext,φins, vins,γimage,γradar,Sss, . . .]
GRAZ UNIVERSITY OF TECHNOLOGY
Advanced Signal Processing Seminar
Current research directions Motion models
Prediction of pedestrian movements Complexity reduction
Employ KF for linear states Parallelization
Distributed PF in a sensor network
Bayesian Methods in Positioning Applications – p.20/21
GRAZ UNIVERSITY OF TECHNOLOGY
Advanced Signal Processing Seminar
Recursive Identification, MIT Press, 1983.
S. Kay Fundamentals of Statistical Signal Processing, Estimation Theory, Prentice Hall PTR, 1993.
P. Djuric et al., Particle Filtering, IEEE Signal Processing Magazine, Vol.20, No.5, September 2003.
F. Gustafsson et al., Particle Filters for Positioning, Navigation and Tracking, IEEE Transactions on Signal Processing, Vol.50, Issue 2, February 2002.
N. Sirola et al., Local Positioning with Parallelepiped Moving Grid, Proceedings of The 3rd Workshop on Positioning, Navigation and Communication (WPNC), 2006.
Bayesian Methods in Positioning Applications – p.21/21
Outline
Bayes Rule
References